Add the following rational expressions. $\dfrac{2k}{k-6}+\dfrac{k^3}{k+9}=$
We can add two rational expressions whose denominators are equal by adding the numerators and keeping the denominator the same. [Does this fit with how we add rational numbers?] When the denominators are not the same, we must manipulate them so that they become the same. In other words, we must find a common denominator. Since the two denominators do not share any common factors, the common denominator is simply the product of these two denominators: $({k-6})\cdot({k+9})$. Let's manipulate the expressions to have that denominator: $\begin{aligned} &\phantom{=}\dfrac{2k}{{k-6}}+\dfrac{k^3}{{k+9}} \\\\ &=\dfrac{2k\cdot({k+9})}{({k-6})\cdot({k+9})}+\dfrac{k^3\cdot({k-6})}{({k+9})\cdot({k-6})} \end{aligned}$ [Why did we do that?] Now that both denominators are the same, let's add! $\begin{aligned} &\phantom{=}\dfrac{2k\cdot(k+9)}{(k-6)\cdot(k+9)}+\dfrac{k^3\cdot(k-6)}{(k+9)\cdot(k-6)} \\\\ &=\dfrac{2k\cdot(k+9)+k^3\cdot(k-6)}{(k-6)(k+9)} \\\\ &=\dfrac{2k^2+18k+k^4-6k^3}{(k-6)(k+9)} \\\\ &=\dfrac{k^4-6k^3+2k^2+18k}{(k-6)(k+9)} \end{aligned}$ In conclusion, $\dfrac{2k}{k-6}+\dfrac{k^3}{k+9}=\dfrac{k^4-6k^3+2k^2+18k}{(k-6)(k+9)}$